Best restricted approximation of smooth function classes

We first discuss the relative Kolmogorov $n$-widths of classes of smooth $2\pi$-periodic functions for which the modulus of continuity of their $r$-th derivatives does not exceed a given modulus of continuity, and then discuss the best restricted approximation of classes of smooth bounded functions defined on the real axis $\mathbb R$ such that the modulus of continuity of their $r$-th derivatives does not exceed a given modulus of continuity by taking the classes of the entire functions of exponential type as approximation tools. Asymptotic results are obtained for these two problems.